
// Make newform 1521.2.a.p in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1521_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_2_a_p();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1521_2_a_p();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1521_a();"
function MakeCharacter_1521_a()
    N := 1521;
    order := 1;
    char_gens := [677, 847];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1521_a_Hecke(Kf)
    return MakeCharacter_1521_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, -1, 0], [0, 0, 0], [-2, 1, -1], [4, -1, 1], [2, -1, 4], [0, 0, 0], [2, 0, -1], [2, 4, -1], [2, -3, 5], [1, 3, -2], [3, -3, -2], [5, 0, 1], [0, -1, 0], [-1, -2, -2], [4, -6, -3], [6, -3, 2], [2, 2, -6], [-4, 2, 3], [1, -1, -3], [-1, 7, -2], [6, -1, -1], [-5, 1, -5], [3, 4, -3], [3, -3, 1], [7, -6, 10], [-4, 9, -3], [-6, 12, -8], [-2, 9, -6], [0, 3, -6], [7, -8, 1], [-9, 4, -10], [-5, -7, -4], [-3, 12, -4], [0, 7, 4], [5, -5, -7], [4, 3, 5], [5, -6, 4], [1, 5, -3], [13, -10, 10], [-7, 5, 0], [-12, -1, -5], [5, 3, 11], [-9, 11, -3], [7, -4, 8], [-23, 6, -9], [-9, 1, 9], [-1, -3, 2], [-6, 13, -8], [11, -4, 14], [5, 1, 2], [0, 0, -13], [7, -4, 11], [1, 5, -3], [2, 0, -1], [3, -5, -11], [19, -7, 9], [6, -6, 17], [1, 8, -13], [10, -11, 4], [-16, -5, 5], [-10, 6, -7], [-16, -1, 9], [3, -21, 8], [1, 11, -3], [20, -10, 11], [-7, -15, 1], [-16, 1, -12], [-1, -11, -10], [11, -16, 15], [-6, -4, 8], [1, -3, -3], [9, -3, -5], [19, -17, 7], [-26, 7, 4], [-11, 10, 7], [5, -5, -7], [15, -4, -6], [8, 4, -11], [9, 11, -4], [4, -13, 19], [-15, -4, -6], [24, -6, 13], [7, 11, 5], [-10, -10, -1], [3, -7, 6], [4, 14, -5], [-8, 22, -15], [-14, 3, 3], [-23, 12, -14], [-24, 20, -28], [-13, 0, 0], [13, 16, -3], [-11, 4, -14], [4, -7, 12], [-26, 16, -6], [-8, 9, -19], [28, -18, 20], [1, 20, 4], [-2, -1, -21], [26, -15, 30], [-1, -20, 9], [-27, 5, 0], [11, 8, 11], [-18, 11, -26], [-16, 19, -17], [-9, 8, -5], [16, 6, 4], [-6, 14, -12], [3, -23, 11], [1, 6, -4], [23, -13, 26], [39, -12, 13], [-10, 12, 12], [-6, 10, -17], [3, 14, -9], [20, 3, -3], [12, -16, 21], [10, -19, 20], [21, -11, 29], [-8, 9, -6], [-5, 7, 5], [-4, 7, -12], [3, -10, -15], [-14, -5, 3], [-6, -13, -8], [-36, 7, 4], [-7, -5, -18], [-28, 8, -28], [-3, 6, 9], [31, -7, 5], [13, -16, -23], [-10, 6, -9], [20, -15, 7], [4, 13, 0], [-13, 19, -13], [20, 0, 18], [-38, 3, 5], [22, -11, 12], [-36, 24, -30], [-17, 26, 0], [16, 10, 11], [2, -5, 20], [18, 25, -7], [-1, -35, 12], [-24, 21, -17], [11, 10, 6], [8, -5, -7], [-20, 14, 8], [-4, -1, -21], [26, 10, -7], [21, -34, 20], [18, 4, 9], [-4, -18, -1], [25, -19, 4], [-28, 3, 8], [5, -1, -8], [6, -5, -14], [21, 2, -16], [-5, 1, -31], [0, -3, -20], [-18, 13, -14], [-13, -9, 5], [-3, -4, -23], [37, 10, 2], [-19, -2, 5], [1, 13, -7], [-20, 9, -6], [-21, -3, 15], [-18, 34, -20], [-11, -30, 7], [-38, 8, 3], [-13, 7, 0], [10, -3, 40], [-17, -10, -9], [9, -18, -27], [-13, 6, -4], [10, -23, 25], [-7, 3, 4], [29, -20, 19], [0, -31, -1], [9, -20, -10], [40, -2, 10], [24, -31, 38], [10, -23, 15], [-26, -10, -3], [-25, 15, -11], [36, 6, 9], [-29, 35, -23], [-14, 38, -7], [15, -24, 33], [15, 3, 24], [-31, 2, -16], [-10, 24, -3], [37, -6, 3], [8, -38, 20], [33, -13, -10], [0, -23, -19], [-12, -6, -8], [25, -22, 38], [-42, -11, 3], [-19, -8, 5], [-11, 16, -15], [-38, 25, -29], [-34, 7, -9], [-19, 2, -11], [-31, -11, 3], [40, 1, 17], [-10, 39, 0], [-12, -3, 10], [21, 6, 10], [9, -6, -12], [-8, 25, 5], [28, -9, 6], [-49, 0, -9], [16, -27, -1], [-24, -6, 24], [-24, 26, -39], [18, -30, 20], [1, 23, -3], [-10, -30, 0], [-44, 18, -12], [21, 21, 6], [25, -1, -17], [-6, 7, -16], [-47, 17, -17], [23, -7, 9], [11, -4, 33], [-9, 37, -3], [-43, -9, 7], [-43, 10, -18], [-32, 37, -45], [-34, 11, 23], [18, -22, 11], [15, 22, -19], [6, 5, -25], [-54, 1, -14], [-13, -32, 13], [-36, 10, -12], [-11, 13, 12], [45, -33, 48], [-4, 7, -14], [19, -26, -3], [37, -2, 10], [-6, 13, -34], [10, 11, 4], [17, 5, -25], [-37, 3, -6], [-35, 21, -54], [32, 22, -13], [-21, 4, -30], [4, 2, -35], [13, -14, -26], [-60, 18, -19], [-35, 4, 3], [-24, 18, -38], [19, -32, 34], [-7, 8, -27], [20, 26, -21], [-5, -16, 28], [13, 8, 31], [-21, 16, 11], [-45, 30, -46], [31, -7, 18], [35, 3, 11], [14, -22, -29], [0, -19, 4], [5, 21, 6], [17, 2, 7], [24, -28, 19], [16, 37, -4], [-34, -23, -2], [-33, 4, 2], [-29, -36, 12], [30, -25, 22], [2, 29, -2], [-26, -12, 26], [-13, 5, 21], [8, -31, 6], [-27, -17, 3], [19, -34, 39], [-41, 9, 20], [-15, 1, -4], [-11, 18, 1], [-32, -13, -8], [38, -26, -6], [-14, -24, -10], [13, -22, 19], [-25, 8, 3], [21, 30, 2], [-45, 3, -29], [-42, 37, -42], [23, -53, 44], [7, -15, -8], [38, -18, -10], [-5, 30, -33], [4, -60, 27], [28, -39, 13], [26, -25, -2], [-28, -4, -19], [25, -1, 32], [-48, 27, -17], [5, 5, 20], [-46, 13, -16], [-1, -15, -3], [-22, -27, 18], [-36, -5, 4], [-15, 4, 32], [-23, 11, 10], [33, 15, -24], [46, -14, -8], [-33, 20, 30], [-27, -25, 16], [34, -20, 9], [-4, 16, -15], [28, 15, -6], [-16, 26, 0], [-16, 37, -9], [53, -18, 16], [3, -25, 30], [4, -7, 12], [2, -17, 20], [-21, 45, -34], [22, -12, -31], [26, -16, 29], [-35, 7, -16], [8, -8, 14], [-14, 26, -6], [-20, 22, 13], [45, 15, 6], [14, 16, -15], [22, 18, 2], [36, 2, -10], [-4, 42, -41], [34, 3, -11], [-14, 55, -10], [-30, -8, -8], [0, 20, -13], [-4, 11, 22], [-30, 42, -35], [-28, 4, -20], [-10, -3, 11], [-52, -7, -4], [-42, 21, -60], [-26, 38, 15], [31, -31, 45], [8, -3, 28], [53, -11, 3], [2, -10, -32], [-39, 29, -16], [55, -2, -17], [27, -51, -9], [-20, -15, 11], [-10, 19, -22], [-18, 45, 10], [44, -24, -26], [19, -55, 41], [-22, -15, 1], [-15, 9, 1], [6, 14, -18], [53, -5, 16], [19, 22, -26], [-28, -21, 30], [8, 15, -10], [22, -19, -1], [23, -47, 37], [-16, -38, 21], [-13, 6, -26], [-27, 22, -10], [-31, -16, -24], [0, 58, -17], [-4, 17, -32], [39, 15, -4], [28, -34, 40], [-23, 15, 42], [18, -29, 15], [-24, 32, 0], [-24, -13, -40], [3, -11, 35], [-51, 30, -46], [37, -13, 14], [37, -20, 2], [-60, 22, -41], [-29, 17, 13], [30, -37, 46], [-47, 49, -67], [24, 33, -10], [50, 12, -2], [-63, 9, -19], [-15, 7, 13], [24, -3, -11], [0, -3, -10], [-34, 2, -15], [-21, 5, 20], [20, -6, -22], [50, -16, 6], [42, -12, 8], [4, 6, -51], [1, -41, 10], [67, 0, -13], [54, -40, 56], [41, 30, -1], [-87, 19, -14], [5, -13, -12], [38, -9, 25], [21, 16, 28], [16, 1, 21], [33, 26, 5], [-10, 51, -40], [10, 4, 52], [-23, -3, -12], [15, -30, 33], [7, 30, -7], [-4, -38, -18], [2, 5, -24], [-9, -35, 16], [2, 31, -12], [5, -5, -20], [41, 0, 52], [-39, 16, -42], [-3, -3, -51], [-5, 63, -26], [29, 23, 11], [-16, -21, 35], [53, -19, 4], [47, -41, -11], [50, -17, 1], [-56, 47, -40], [8, -45, 47], [-75, 1, 17], [52, -20, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1521_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_2_a_p();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1521_2_a_p(:prec:=3)
    chi := MakeCharacter_1521_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1521_2_a_p();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1521_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1521_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -2, 1, 1]>,<5,R![-1, 3, 4, 1]>,<7,R![-29, 31, -10, 1]>],Snew);
    return Vf;
end function;